The Deborah Number Dilemma

Research groups around the world are working to determine which theoretical models describing polymer flows most closely match experimental findings. Over the past 15 to 20 years, dozens of models have been proposed, but they could be tested only under limited conditions. "As a consequence," says Leal, "nobody really had a good handle on which, if any, of these equations describe the behavior of real materials in more complex flows typical of manufacturing systems."

A major stumbling block has been a computational limitation associated with the Biblical prophet Deborah: She said the mountains flow according to the Lord's time scale and not humans. In the case of polymers, a high Deborah number means the flow is strong enough that the polymer becomes highly oriented in one direction and stretched. For most flows, this occurs when the time it takes for the polymer to relax is long compared to the rate at which the flow is deforming it. Fluids that act this way are called "non-Newtonian." Unlike Newtonian fluids such as water, when they're stretched they don't immediately return to their unstressed state.

"When computational studies of polymer flow began 15 to 20 years ago," explains Leal, "researchers found that numerical methods used for Newtonian fluids wouldn't work for any flow beyond a Deborah number of approximately one. Not only are the equations more complicated, with more unknowns, but they couldn't be solved for any flow in a Deborah number range of technological significance. Gradually, driven by the work of researchers worldwide and a series of international workshops, the so-called high Deborah number problem has been whittled away. Most of the issues are resolved, and we're at a point where we can compute nontrivial flows."